## Root mean squared radius of WS

The Woods-Saxon potential is

$f(r) = 1/(1+Exp(\frac{r-R}{a}))$

where $R=r_0 A^{1/3}$ is the half-maximum radius that $r_0$ is the reduced radius and $A$ is the nuclear mass number, and $a$ is the diffusiveness parameter.

$\sqrt{\left} = \sqrt{\int{f(r) r^2 r^2 dr} / \int{f(r) r^2 dr}}$

where the “extract” $r^2$ is because of spherical coordinate.

The integration is a polynomial

$\int{f(r) r^n dr} = -a^{n+1} \Gamma(n+1) \sum_{k=1}^{\infty} \frac{(-Exp(R/a))^k}{k^{n+1}}$

In mathematica, the sum is notated by,

$PolyLog(n, x) = \sum_{k=1}^{\infty} \frac{x^k}{k^n}$

Thus, the rms radius for $a > 0$ is

$\sqrt{\left} = a \sqrt{12 \frac{PolyLog(5, -Exp(R/a))}{PolyLog(3, -Exp(R/a))} }$

For $a = 0$, $f(r) = 1, r < R$,

$\sqrt{\left} = \sqrt{\frac{3}{5}}R$

## Woods-Saxon Shape

this is a collective model of the nuclei density vs radius. it has another name Fermi-shape.

$\rho(r) = \frac { \rho_0} {1+ Exp( (r - R_0)/a) }$

where $\rho_0$ is central density, or density at r = 0. $R_o$ is the radius of half density and $a$ is the diffuseness. when a is large, the “tail” of the shape will be longer.

the radius is measured in unit of fm $1 fm = 10^{-15} m$.

$R_0$ can be vary from 1 fm to 7 or 8 fm.  for 16O, it is about 2 fm. and for 208Pb, it is about 6fm .

$a$ is more or less the same for different nuclei.

the density of nuclei can be mass density or charge density, the Woods-Saxon also gives a good approximation.

for mass density, a $10^{17} kg m^{-3}$. for comparison, water density is 1 kg per meter cube.

and charge density is about $0.15 e fm^{-3}$.